
By (REF ), (REF ) and the chain rule of differentiation
[1]},
\(\frac{\partial F}{\partial \gamma }&={\sf Tr}\left\lbrace \frac{\partial F}{\partial \mathbf {D}}\, \frac{\partial \mathbf {D}}{\partial \gamma }\right\rbrace \\&=\frac{|\mathbf {D}|^{\frac{2}{n}}}{n}{\sf Tr}\Biggl \lbrace \left[\exp \left[\frac{2}{n}I(\mathbf {Z};\mathbf {D}\mathbf {X}+\mathbf {Z})\right]\mathbf {N}^{-1}{\sf Cov}(\mathbf {Z}|\mathbf {D}\mathbf {X}+\mathbf {Z})- |\mathbf {I}-\mathbf {D}^{-2}|^{\frac{1}{n}}\left[\mathbf {I}+(\mathbf {D}^{2}-\mathbf {I})^{-1}\right]\right]\frac{\mathbf {I}-\mathbf {D}^{-2}}{\gamma }\Biggr \rbrace \\&=\frac{|\mathbf {D}|^{\frac{2}{n}}}{n\gamma }{\sf Tr}\left\lbrace \exp \left[\frac{2}{n}I(\mathbf {Z};\mathbf {D}\mathbf {X}+\mathbf {Z})\right]\mathbf {N}^{-1}{\sf Cov}(\mathbf {Z}|\mathbf {D}\mathbf {X}+\mathbf {Z})(\mathbf {I}-\mathbf {D}^{-2})- |\mathbf {I}-\mathbf {D}^{-2}|^{\frac{1}{n}} \mathbf {I}\right\rbrace  \\&=\frac{|\mathbf {D}|^{\frac{2}{n}}}{n\gamma }\left\lbrace \exp \left[\frac{2}{n}I(\mathbf {Z};\mathbf {D}\mathbf {X}+\mathbf {Z})\right]{\sf Tr}\left\lbrace \mathbf {N}^{-1}{\sf Cov}(\mathbf {Z}|\mathbf {D}\mathbf {X}+\mathbf {Z})(\mathbf {I}-\mathbf {D}^{-2})\right\rbrace -n|\mathbf {I}-\mathbf {D}^{-2}|^{\frac{1}{n}}\right\rbrace .\) 
